Induced EMF in a coil outside a solenoid

[cmb323]

Homework Statement

[/B]
A conducting coil of radius R is outside a long solenoid with a cross section of radius r. What is the emf induced in the coil?

Most example problems of this type I think are solved based on Faraday’s Law. These examples do not use the distance from the solenoid to the coil (simplifying assumption?). Surely this distance is important?

I can derive a formula to find the electric field at point R. Can this be used to calculate the emf induced in the coil by the solenoid?

Homework Equations

The Attempt at a Solution

 

[berkeman]

Homework Statement

[/B]
A conducting coil of radius R is outside a long solenoid with a cross section of radius r. What is the emf induced in the coil?

Most example problems of this type I think are solved based on Faraday’s Law. These examples do not use the distance from the solenoid to the coil (simplifying assumption?). Surely this distance is important?

I can derive a formula to find the electric field at point R. Can this be used to calculate the emf induced in the coil by the solenoid?

Homework Equations

The Attempt at a Solution

Welcome to the PF.

Can you show us a diagram from a typical problem like this? And what explicit equations would you write for the diagram that you will post? Thanks.

 

[cmb323]

I can’t provide a diagram. However, the solenoid and conducting coil are on the same axis. As a result the magnetic field lines inside the solenoid are perpendicular to the face of conducting coil.

Hopefully not too many typos in the following.

Outside the solenoid R > r

E = ((µ0nr2)/(2R))di/dt

A common equation used to find the induced emf in the exterior ring is

Emf = (µ0nπr2)(di/dt) but R is not in the equation?

 

[rude man]

This problem is oh so badly defined. What's the solenoid current? Number of turns? Is the coil near the middle of the solenoid or away from an end? How far if so? Etc.
As for the reason R doesn't appear, that's because it's immaterial.
As berkeman says, you need to provide a picture. If you can't do that you don't understand the problem you're supposed to be addressing.

 

[cmb323]

I can “plug and chug” into the equations and answer the question, but it is the “why” that concerns me. As a consequence the current, number of turns, etc. does not change the question. Your statement “…R doesn’t appear, because it is immaterial” is the question. I’m struggling to believe that if R is 1 cm or 1 km it makes no difference?

 

[rude man]

I can “plug and chug” into the equations and answer the question, but it is the “why” that concerns me. As a consequence the current, number of turns, etc. does not change the question. Your statement “…R doesn’t appear, because it is immaterial” is the question. I’m struggling to believe that if R is 1 cm or 1 km it makes no difference?

OK, I will assume that the coil is roughly half-way between the solenoid ends and concentric with same. (If the coil is beyond both ends of the solenoid the flux penetrating the coil will be very much smaller.)

What is Farady's law? That will answer your question about R being immaterial.

 

[cmb323]

Thank you. I see how Faraday’s Law would work inside of the solenoid. However, there is no magnetic field outside the solenoid so the electrons in the external ring have no clue that the solenoid exists. Therefore is seems wrong to use the following equation to find the emf in the ring

Emf = (µ0nπr2)(di/dt)

I recognize that the changing magnetic field within the solenoid also induces an electric field which I assume exists outside of the solenoid. Can this induced electric field be used to calculate the induced emf in the ring?

 

[rude man]

Thank you. I see how Faraday’s Law would work inside of the solenoid. However, there is no magnetic field outside the solenoid so the electrons in the external ring have no clue that the solenoid exists.

Apply faraday's law to the ring! There IS mag flux within the ring because the ring sits within the ends of the solenoid (again, my assumption; if the ring sits beyond both solenoid ends the story is different. Please confirm which is the case!

Therefore is seems wrong to use the following equation to find the emf in the ring
Emf = (µ0nπr2)(di/dt)

This is not farady's law. And your formula applies to the solenoid emf, not that of the ring.

I recognize that the changing magnetic field within the solenoid also induces an electric field which I assume exists outside of the solenoid. Can this induced electric field be used to calculate the induced emf in the ring?

No, has nothing to do directly with any electric field anywhere.

 

[cmb323]

The ring is far from the ends of the solenoid, and I agree that there is a changing flux enclosed by the ring. The problem (in a famous textbook) that caused me to question the validity of the emf equation had r = 0.01 m and R = 0.02 m. Since r is close to R perhaps a simplifying assumption that is not stated or just an erroneous use of the formula?

If R is increased to 1 km, then the equation (does not include R) gives the same induced emf – this fails any test of reasonableness.

I think that the solution for emf induced in the ring is above my pay grade and requires the use of magnetic vector potential. A possible equation along these lines has R in the denominator, thereby passing a test of reasonableness.

 

[rude man]

The ring is far from the ends of the solenoid, and I agree that there is a changing flux enclosed by the ring. The problem (in a famous textbook) that caused me to question the validity of the emf equation had r = 0.01 m and R = 0.02 m. Since r is close to R perhaps a simplifying assumption that is not stated or just an erroneous use of the formula?

If R is increased to 1 km, then the equation (does not include R) gives the same induced emf – this fails any test of reasonableness.

I think that the solution for emf induced in the ring is above my pay grade and requires the use of magnetic vector potential. A possible equation along these lines has R in the denominator, thereby passing a test of reasonableness.

What equation would that be?

No, it's not 'above your pay grade' and no it does not require magnetic vector potential. So cheer up!

Ok, to review: I am assuming the ring is (1) coaxial with the solenoid and (2) it is located somewhere near the middle of the solenoid lengthwise or at least not very close to either end.

Bearing this in mind, I know that it sounds incredible that the induced emf is the same irrespective of the diameter of the ring, but such is the case and it can easily be proven by one of Maxwell's equations together with the Stokes theorem. The line integral of the E field of a given arc length will be smaller the larger R is. But the line integral of E around the entire ring will still be the same irrespective of the size of R.

I do want to mention one caveat: Faraday says that emf = - ∂φ/∂t but it must be appreciated that φ is the TOTAL flux including any set up by the current-carrying ring itself. So in empty space it's exact but with finite current in the ring the total emf changes to - (∂φ_e/∂t - ∂φ_r/∂t) with φ_e the externally applied flux (i.e. the solenoid in your case) and φ_r the flux set up by the current-carrying ring. Usually it's assumed that the current is small enough so the second term can be ignored.

And even that statement is approximate. The complete solution must include the ring-induced flux modifying the solenoid flux, since ring flux couples into the solenoid just as solenoid flux couples into the ring. But we won't go there this time.

 

[cmb323]

Great explanation, but I think I went straight to the deep end – thanks for the life preserver.

The equation that was used to find the emf in the problem was µ0nπr2(di/dt). So from what you have stated I assume this is was a correct application?

 

[rude man]

Great explanation, but I think I went straight to the deep end – thanks for the life preserver.

The equation that was used to find the emf in the problem was µ0nπr2(di/dt). So from what you have stated I assume this is was a correct application?

I'm assuming your equation is really µ₀nπr²(di/dt). That would be the emf induced across the solenoid. But that emf is N dφ/dt where N = no of turns of the solenoid and φ is the solenoid flux. But that's the same flux coupling the ring so what must be the emf around the ring? (Hint, maybe: this is an N:1 step-down transformer with the secondary loaded by the ring resistance.)

 

[cmb323]

The life preserver just was pulled out of my reach – transformers are yet to come in my tortuous journey through electromagnetism. With regard to the flux I believe you saying that the flux through the solenoid is the same as the flux through the ring – agree. Then despite my misgivings then µ0nπr2(di/dt) musr be good for all R?

 

[rude man]

The life preserver just was pulled out of my reach – transformers are yet to come in my tortuous journey through electromagnetism. With regard to the flux I believe you saying that the flux through the solenoid is the same as the flux through the ring – agree. Then despite my misgivings then µ0nπr2(di/dt) musr be good for all R?

No. The flux is the same for both the solenoid & ring but the ring has only 1 winding whereas your solenoid has nL windings, L being its effective length - and Faraday says what about induced emf?

 

[cmb323]

The change in flux in Faradays Law is on a per coil basis; therefore the rate of change of the flux with respect to time is preceded by N – the number of turns on the solenoid. To get the answer to the problem in the text n in the equation needs to be the number of turns on the solenoid. However on my journey I have become skeptical of “right” being defined by matching the answer in the back of the book.

 

[rude man]

The change in flux in Faradays Law is on a per coil basis; therefore the rate of change of the flux with respect to time is preceded by N – the number of turns on the solenoid. To get the answer to the problem in the text n in the equation needs to be the number of turns on the solenoid. However on my journey I have become skeptical of “right” being defined by matching the answer in the back of the book.

And not without cause! We here at PF have over time found numerous wrong answers in textbooks as well as those given by the instructor.

So - you know the flux in the solenoid, you know that equals the flux in the ring, you know Faraday's law for emf induced in the ring by a time-varying flux - so that gives you your answer.

The formula you gave for flux in the solenoid is at least dimensionally correct (I haven't checked it myself) so there should not be an issue with n being no. of turns per meter vs. no. of turns, or any other dimensional discrepancy.